On the cadlaguity of random measures. (English) Zbl 0542.60050

Adler, Robert J.; Feigin, Paul D.

Ann. Probab. 12, 615-630 (1984).

Let \(X=\{X(B):B\in {\mathcal B}^ k\}\) be a stochastically continuous, finitely additive but not necessarily positive random measure, on the Borel \(\sigma\)-algebra \({\mathcal B}^ k\) in \(I_ k=[0,1]^ k\), with independent increments: \(X(B_ 1),...,X(B_ n)\) are independent for all choices of n and disjoint \(B_ 1,...,B_ n\). In addition, let \(\nu\), defined on \(B^ k\times {\mathbb{R}}\), be the Lévy measure of X, i.e., neglecting deterministic and Gaussian components, \[ E[e^{iuX(B)}]=\exp [\int_{| x|>1}(e^{iuy}-1)\nu(B,dx)+\int_{| x| \leq 1}(e^{iuy}-1-iuy)\nu(B,dx)]. \] The authors examine the question of whether, given a family \({\mathcal A}\) of closed subsets of \(I_ k\), there exists a version of X that is càdlàg on \({\mathcal A}\) in the sense that almost surely, for every decreasing sequence \((A_ n)\) in \({\mathcal A}\) and \(A\in {\mathcal A}\) with \(\lambda(A_ n\backslash A)\to 0 (\lambda =Lebesgue\) measure), \(X(A_ n)\to X(A)\), while for every increasing sequence \((A_ n)\) in \({\mathcal A}\), lim X\((A{}_ n)\) exists and is finite. Their main results are that the answer is always affirmative provided that \(\int^{1}_{-1}| x| \nu(I_ k,dx)<\infty\), but that when this integral diverges restriction on the size of A is required.
In particular, a sufficient condition for existence of a version of X that is càdlàg on A is convergence of a certain series involving metric entropy with inclusion of A and some integrals with respect to \(\nu\). Related results of R. F. Bass and R. Pyke [Z. Wahrscheinlichkeitstheor. Verw. Geb. 66, 157-172 (1984; Zbl 0525.60079)] and a counterexample in this paper indicate that this metric entropy condition is essentially the best possible, at least for stable measures.

Reviewer: A.F.Karr

MSC:

60G57	Random measures
60G17	Sample path properties
60J99	Markov processes
60G15	Gaussian processes

Keywords:

cadlag process; Lévy measure; metric entropy condition; stable measures

Citations:

Zbl 0525.60079